CBSE 10th Mathematics | Introduction of Polynomials

POLYNOMIALS

An expression $\displaystyle p(x)$ of the form $\displaystyle p(x)={{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+...+{{a}_{n}}$ where all $\displaystyle {{a}_{0}},\,\,{{a}_{1}},\,\,{{a}_{2}},\,....,\,\,{{a}_{n}}$ are real numbers and n is a non-negative integer, is called a polynomial.

The degree of a polynomial in one variable is the greatest exponent of that variable.

$\displaystyle {{a}_{0}},\,\,{{a}_{1}},\,\,{{a}_{2}}\,...,\,\,{{a}_{n}}$ are called the co-efficients of the polynomial $\displaystyle p(x)$ .

an is called constant term.

DEGREE OF A POLYNOMIAL

The exponent of the term with the highest power in a polynomial is known as its degree.

$\displaystyle f(x)=8{{x}^{3}}-2{{x}^{2}}+8x-21$ and $\displaystyle g(x)=9{{x}^{2}}-3x+12$ are polynomials of degree 3 and 2 respectively.

Thus, $\displaystyle f(x)={{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+\ldots +{{a}_{n-1}}x+{{a}_{n}}$ is a polynomial of degree n, if $\displaystyle {{a}_{0}}\ne 0.$

On the basis of degree of a polynomial, we have following standard names for the polynomials.

A polynomial of degree 1 is called a linear polynomial. Example: $\displaystyle 2x+3,\,\,\frac{1}{3}u+7$ etc.

A polynomial of degree 2 is called a quadratic polynomial. Example: $\displaystyle {{x}^{2}}+2x+3$ , $\displaystyle {{y}^{2}}-9$ etc.

A polynomial of degree 3 is called a cubic polynomial. Example: $\displaystyle {{x}^{3}}+7x-3$ , $\displaystyle -{{x}^{3}}+{{x}^{2}}+\sqrt{3}x$ etc.

A polynomial of degree 4 is called a biquadratic polynomial. Example: $\displaystyle 3{{u}^{4}}-5{{u}^{3}}+2{{u}^{2}}+7$ .

VALUE OF A POLYNOMIAL

If $\displaystyle f(x)$ is a polynomial and a is any real number, then the real number obtained by replacing x by a in $\displaystyle f(x)$ is called the value of $\displaystyle f(x)$ at x = a and is denoted by $\displaystyle f(\alpha )$ .

e.g. : Value of $\displaystyle p(x)=5{{x}^{2}}-3x+7$ at $\displaystyle x=1$ will be

\ $\displaystyle p(1)=5{{(1)}^{2}}-3(1)+7$

$\displaystyle =5-3+7=9$

ZEROS OF A POLYNOMIAL

A real number a is a zero of polynomial $\displaystyle f(x)$ if $\displaystyle f(\alpha )$ = 0.

The zero of a linear polynomial $\displaystyle ax+b$ is $\displaystyle -\frac{b}{a}$ . i.e. $\displaystyle -\frac{Constant\,\text{term}}{\text{Coefficient of }x}$

Geometrically zero of a polynomial is the point where the graph of the function cuts or touches x-axis.

When the graph of the polynomial does not meet the x-axis at all, the polynomial has no real zero.

SIGNS OF COEFFICIENTS OF A QUADRATIC POLYNOMIAL

The graphs of $\displaystyle y=a{{x}^{2}}+bx+c$ are given in figure. Identify the signs of a, b and c in each of the following:

(i) We observe that $\displaystyle y=a{{x}^{2}}+bx+c$ represents a parabola opening downwards. Therefore, a < 0. We observe that the turning point $\displaystyle \left( -\frac{b}{2a},\,-\frac{D}{4a} \right)$ of the parabola is in first quadrant where $\displaystyle D={{b}^{2}}-4ac$

\ $\displaystyle -\frac{b}{2a}>0$ Þ $\displaystyle -b<0$ Þ $\displaystyle b>0$ $\displaystyle [\because \,a<0]$

Parabola $\displaystyle y=a{{x}^{2}}+bx+c$ cuts y-axis at Q. On y-axis, we have x = 0. Putting x = 0 in $\displaystyle y=a{{x}^{2}}+bx+c,$ we get $\displaystyle y=c.$

So, the coordinates of Q are (0, c). As Q lies on the positive direction of y-axis. Therefore, c > 0.

Hence, $\displaystyle a<0,\,\,b>0$ and $\displaystyle c>0.$

(ii) We find that $\displaystyle y=a{{x}^{2}}+bx+c$ represents a parabola opening upwards. Therefore, $\displaystyle a>0.$ The turning point of the parabola is in fourth quadrant.

\ $\displaystyle \frac{-b}{2a}>0\Rightarrow -b>0\Rightarrow b<0.$ $\displaystyle [\because \,a>0]$

Parabola $\displaystyle y=a{{x}^{2}}+bx+c$ cuts y-axis at Q and y-axis. We have x = 0. Therefore, on putting x = 0 in $\displaystyle y=a{{x}^{2}}+bx+c,$ we get $\displaystyle y=c.$

So, the coordinates of Q are (0, c). As Q lies on negative y-axis. Therefore, c < 0.

Hence, a > 0, b < 0 and c < 0.

(iii) Clearly, $\displaystyle y=a{{x}^{2}}+bx+c$ represents a parabola opening upwards.

Therefore, a > 0. The turning point of the parabola lies on OX.

\ $\displaystyle -\frac{b}{2a}>0\Rightarrow -b>0\Rightarrow b<0$ $\displaystyle [\because \,a>0]$

The parabola $\displaystyle y=a{{x}^{2}}+bx+c$ cuts y-axis at Q which lies on positive y-axis. Putting
x = 0 in $\displaystyle y=a{{x}^{2}}+bx+c,$ we get y = c. So, the coordinates of Q are (0, c). Clearly, Q lies on OY.

\ c > 0.

Hence, a > 0, b < 0, and c > 0.

(iv) The parabola $\displaystyle y=a{{x}^{2}}+bx+c$ opens downwards. Therefore, a < 0.

The turning point $\displaystyle \left( -\frac{b}{2a},-\frac{D}{4a} \right)$ of the parabola is on negative x-axis,

\ $\displaystyle -\frac{b}{2a}<0\,\,\Rightarrow \,\,\,b<0$ $\displaystyle [\because \,a<0]$

Parabola $\displaystyle y=a{{x}^{2}}+bx+c$ cuts y-axis at Q (0, c) which lies on negative y-axis. Therefore, c < 0.

Hence, $\displaystyle a<0,\,\,b<0$ and $\displaystyle c<0.$

(v) We notice that the parabola $\displaystyle y=a{{x}^{2}}+bx+c$ opens upwards. Therefore, a > 0.

Turning point $\displaystyle \left( -\frac{b}{2a},-\frac{D}{4a} \right)$ of the parabola lies in the first quadrant.

\ $\displaystyle -\frac{b}{2a}>0\,\,\,\Rightarrow \,\,\,\frac{b}{2a}<0\,\,\,\,\Rightarrow b<0$ $\displaystyle [\because \,a>0]$

As Q (0, c) lies on positive y-axis. Therefore, c > 0.

Hence, $\displaystyle a>0,\,\,b<0$ and c > 0.

(vi) Clearly, $\displaystyle a<0\,\,$

$\displaystyle \,Q\left( -\frac{b}{2a},-\frac{D}{4a} \right)$ lies in the fourth quadrant.

\ $\displaystyle -\frac{b}{2a}>0\,\,\,\Rightarrow \,\,\,\frac{b}{2a}<0\,\,\,\Rightarrow \,\,\,b>0$ $\displaystyle [\because \,a<0]$

As Q (0, c) lies on negative y-axis. Therefore,
c < 0.

Hence, $\displaystyle a<0,\,\,b>0$ and c < 0.

Question:

The graphs of $\displaystyle y=p(x)$ are given below for some polynomial $\displaystyle p(x)$ . Find the number of zeros of $\displaystyle p(x)$ in each case.

Solution:

(i) The polynomial represented in (i) has no zero because its graph does not intersect x-axis at any point.

(ii) The polynomial represented in (ii) has one zero because its graph intersects x-axis at one point.

(iii) The polynomial represented in (iii) has three zeros because its graph intersects x-axis at three points.

(iv) The polynomial represented in (iv) has two zeros because its graph intersect x-axis at two points.

(v) The polynomial represented in (v) has four zeros because its graph intersect x-axis at four points.

(vi) The polynomial represented in (vi) has three zeros because its graph intersects x-axis at three points.

CBSE Tutorials

CBSE 10th Mathematics | Introduction of Polynomials

POLYNOMIALS

DEGREE OF A POLYNOMIAL

VALUE OF A POLYNOMIAL

ZEROS OF A POLYNOMIAL

SIGNS OF COEFFICIENTS OF A QUADRATIC POLYNOMIAL

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CBSE 10th Mathematics | Introduction of Polynomials

POLYNOMIALS

DEGREE OF A POLYNOMIAL

VALUE OF A POLYNOMIAL

ZEROS OF A POLYNOMIAL

SIGNS OF COEFFICIENTS OF A QUADRATIC POLYNOMIAL

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Solution:

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